
We assumed that solitons were uniformly distributed in space. This is appropriate for the thermodynamics, and is what happens in the periodic setting in the KdV spectral theory. In the more general, quasi-periodic setting of [1]}, [2]} the DOS is given by \(f(\eta )= \varphi (\eta )\varkappa (\eta )\) , where \(\varkappa (\eta )\)  is the function responsible for the variable spatial density of solitons depending on their spectral parameter (in the periodic setting \(\varkappa (\eta )=1/L\) ).

The formula (REF ), as derived, is, strictly speaking, valid only for a rarefied gas. However, as mentioned, its validity in the context of a dense gas may be argued for by the idea that the effect of the \(i^{\rm th}\)  soliton is indeed restricted to where the \(i^{\rm th}\)  quasi-particle lies. Full justification is obtained by the thermodynamic finite-gap derivation of [1]} (see also [4]} and the discussion in Section REF ).

The classical interpretation of \(\mathcal {K}(\eta )\)  in (REF ) as a density of asymptotic space was first proposed in [5]} in the context of a geometric, semi-classical intepretation of the GHD equations. The metric in asymptotic space, \({\rm d}x^-(\eta )^2 = g(\eta ) {\rm d}x^2\) , as viewed by particle \(\eta \) , is \(g(\eta ) = \mathcal {K}(\eta )^2\)  according to (REF ). Interestingly, it relates the classical concept of a metric in asymptotic space, with the quantum Bethe ansatz concept of the total density \(\rho _{\rm s}(\eta )\) . In the soliton gas theory, it gives a geometric interpretation of the quantity \(\sigma (\eta )\) .

